Question

Solve for x .

Want steps also

Answer 1

NOTE: I am not adding log 10 as the base. Because it's very difficult to write here.Assume that there is base 10 to all the logs.

Given Equation is :

$x^{logx} = 100x$

$log( x^{logx}) = log(100x)$

log(x)log(x) = log(100 * x)   ------------ (1)

RHS:

We know that log(ab) = log a + log b

Now,

log(100 * x) = log(x) + log(100)

                    = log x + log 2    ----------- (2)


Substitute (2) in (1), we get

log(x) * log(x) = log(x) + log (2)

Let us assume log x as x.

x * x = x + 2

x^2 = x + 2

x^2 - x - 2 = 0

x^2 - 2x + x - 2 = 0

x(x - 2) + 1(x - 2) = 0

(x - 2)(x + 1) = 0

x = 2 (or) x = -1.

So, 

log (x) = 2

      x = 100.


log(x) = -1

     x = 1/10.


Therefore x = 100 (or) x = 1/10.


Hope this helps!

Answer 2

log( x^{logx}) = log(100x)log(x​logx​​)=log(100x) 

------------------

log(x)log(x) = log(100 * x)   -

RHS:

======== log(ab) = log a + log b


log(100 * x) = log(x) + log(100)

                    = log x + log 2    -----


--------. from (2) in (1), we get

log(x) * log(x) = log(x) + log (2)

Let us assume log x as x.

x * x = x + 2

x^2 = x + 2

x^2 - x - 2 = 0

x^2 - 2x + x - 2 = 0

x(x - 2) + 1(x - 2) = 0

(x - 2)(x + 1) = 0

x = 2 (or) x = -1.


________________________

log (x) = 2

      x = 100.


log(x) = -1

     x = 1/10.


Therefore x = 100
/ x = 1/10.